Hy‑CycleAlign: Hyperbolic Registration Fusion

• The paper introduces a dual-path cyclic framework that leverages hyperbolic space to align misaligned infrared and visible images.
• It employs a Hyperbolic Hierarchy Contrastive Alignment module to impose pixel-level and edge-level constraints for enhanced gradient sensitivity.
• Improved registration quality directly enhances fusion outcomes, demonstrating competitive performance on benchmarks like DroneVehicle and MFNet.

Hyperbolic Cycle Alignment Network (Hy‑CycleAlign) is a registration–fusion framework for infrared–visible image fusion under misalignment. It formulates cross-modal image registration in hyperbolic space through a dual-path cyclic architecture: a forward registration network aligns infrared to visible, a backward registration network reconstructs the original image, and a Hyperbolic Hierarchy Contrastive Alignment module imposes pixel-level and edge-level constraints in the Poincaré ball. The method is presented as the first image registration method based on hyperbolic space, and its stated objective is to improve both alignment quality and downstream fusion quality under nonlinear cross-modal discrepancies (Li et al., 31 Jul 2025).

1. Problem setting and formal task

Hy‑CycleAlign addresses infrared–visible image fusion when the two input modalities are spatially misaligned. The motivating setting is that infrared and visible images often exhibit sensor-pose differences, viewpoint changes, and modality gaps induced by distinct imaging physics, while most existing fusion methods assume pre-aligned inputs. Registration methods that do exist are described as typically working in Euclidean space and relying on image translation modules, which can accumulate error and introduce artifacts (Li et al., 31 Jul 2025).

The framework uses infrared images $T$ and visible images $V$ as inputs. Two registration networks are defined: $R_{t2v}: T \to V$ for infrared-to-visible alignment and $R_{v2t}: V \to T$ for visible-to-infrared alignment. The forward path predicts a deformation field $\phi_{t2v} = R_{t2v}(T,V)$ and warps the infrared image as

$T_v = T \circ \phi_{t2v}.$

The backward path predicts $\phi_{v2t} = R_{v2t}(T,V)$ and produces

$V_t = V \circ \phi_{v2t}.$

These transformations are then composed to reconstruct the original modalities:

$$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$

This establishes the closed-loop registration structure that gives the method its “cycle alignment” designation (Li et al., 31 Jul 2025).

Hy‑CycleAlign also includes a fusion stage. After registration, the aligned infrared image $T_v$ and the visible image $V$0 are passed to a simple encoder–decoder fusion network:

$V$1

This design makes registration quality a direct determinant of fusion quality rather than a preprocessing step external to fusion (Li et al., 31 Jul 2025).

2. Hyperbolic geometric rationale

The method is motivated by the claim that Euclidean geometry handles linear discrepancies well but is limited in representing complex nonlinear relationships between infrared and visible imagery. The paper attributes this to the linear nature of Euclidean distances and the difficulty of capturing hierarchical, nonlinear, and modality-specific correspondences with standard translation-based alignment (Li et al., 31 Jul 2025).

Hy‑CycleAlign instead uses the Poincaré ball model of hyperbolic space,

$V$2

and compares Euclidean and hyperbolic sensitivities to misalignment. The Euclidean distance is written as

$V$3

The Poincaré distance is given as

$V$4

Under the approximation $V$5, with $V$6, the paper derives

$V$7

This yields the gradient amplification relation

$V$8

As $V$9, the paper states that $R_{t2v}: T \to V$0, implying that small misalignments generate stronger gradients in hyperbolic space, especially near the boundary of the ball (Li et al., 31 Jul 2025).

The geometric interpretation is that hyperbolic space is more sensitive to misalignment than Euclidean space and that this sensitivity increases as points approach the boundary. In the method’s argument, this sensitivity makes hyperbolic constraints particularly suitable for cross-modal registration, where subtle structural discrepancies between thermal and visible cues are often decisive (Li et al., 31 Jul 2025).

3. Dual-path cross-modal cyclic registration

The registration backbone is organized as a dual-path cycle. The forward registration network aligns infrared to visible, while the backward registration network aligns visible to infrared. Their outputs are not treated as independent deformations; instead, they are composed to form closed registration loops, producing $R_{t2v}: T \to V$1 and $R_{t2v}: T \to V$2, which are compared against the original $R_{t2v}: T \to V$3 and $R_{t2v}: T \to V$4. This establishes geometric consistency through inverse-like behavior without requiring explicit inverse parameterization (Li et al., 31 Jul 2025).

The cycle consistency term is

$R_{t2v}: T \to V$5

A smoothness regularizer is applied to the set of deformation fields $R_{t2v}: T \to V$6:

$R_{t2v}: T \to V$7

The smoothness term encourages plausible spatial transformations, while the cycle term penalizes inconsistent forward–backward warps (Li et al., 31 Jul 2025).

Structural realism is further enforced through adversarial supervision on Sobel edge maps. For infrared-to-visible alignment, the adversarial objective is written as

$R_{t2v}: T \to V$8

A symmetric term is used for visible-to-infrared registration with a second discriminator $R_{t2v}: T \to V$9. The discriminators therefore judge whether edges from the registered image resemble edges from the target modality, which ties registration to cross-modal structural plausibility rather than only pixelwise similarity (Li et al., 31 Jul 2025).

This cyclic design also differentiates Hy‑CycleAlign from alignment-free fusion frameworks. Its architecture couples registration, structural consistency, and fusion, rather than assuming that fusion can compensate for input misalignment on its own (Li et al., 31 Jul 2025).

4. Hyperbolic Hierarchy Contrastive Alignment

The Hyperbolic Hierarchy Contrastive Alignment module, abbreviated H$R_{v2t}: V \to T$0CA, is the method’s central hyperbolic component. It operates at two levels: H$R_{v2t}: V \to T$1CA‑p for pixel or deep feature alignment and H$R_{v2t}: V \to T$2CA‑e for edge-level alignment. The “hierarchy” in the module refers to this multi-level combination of global feature alignment and local structural alignment (Li et al., 31 Jul 2025).

Euclidean features are mapped into the Poincaré ball using

$R_{v2t}: V \to T$3

and, in the appendix formulation, the map is accompanied by a boundary projection

$R_{v2t}: V \to T$4

with $R_{v2t}: V \to T$5. The paper uses curvature $R_{v2t}: V \to T$6 in the reported model configuration (Li et al., 31 Jul 2025).

Within the ball, Hy‑CycleAlign uses Möbius addition

$R_{v2t}: V \to T$7

and the hyperbolic distance

$R_{v2t}: V \to T$8

These operations replace Euclidean addition and Euclidean distances in the alignment loss (Li et al., 31 Jul 2025).

At the edge level, H$R_{v2t}: V \to T$9CA‑e uses Sobel operator outputs and minimizes

$\phi_{t2v} = R_{t2v}(T,V)$0

At the pixel or deep-feature level, H$\phi_{t2v} = R_{t2v}(T,V)$1CA‑p uses VGG‑16 features and minimizes

$\phi_{t2v} = R_{t2v}(T,V)$2

The full hyperbolic contrastive term is

$\phi_{t2v} = R_{t2v}(T,V)$3

In this construction, H$\phi_{t2v} = R_{t2v}(T,V)$4CA‑e targets local structural correspondence, while H$\phi_{t2v} = R_{t2v}(T,V)$5CA‑p targets higher-level and more global feature correspondence, both under hyperbolic geometry (Li et al., 31 Jul 2025).

A common misconception is that the hyperbolic component alone performs registration. The reported formulation does not support that reading: H$\phi_{t2v} = R_{t2v}(T,V)$6CA is one component inside a broader cycle-consistent registration system that also uses adversarial edge supervision, smooth deformation regularization, and a fusion loss (Li et al., 31 Jul 2025).

5. Objective function and training protocol

The overall objective combines adversarial, cycle, hyperbolic, smoothness, and fusion terms:

$\phi_{t2v} = R_{t2v}(T,V)$7

The fusion term is

$\phi_{t2v} = R_{t2v}(T,V)$8

Its first term encourages preservation of salient intensities, while its second term encourages preservation of the strongest gradients from either modality (Li et al., 31 Jul 2025).

Training is performed jointly over the two registration networks, the two discriminators, the hyperbolic alignment module, and the fusion network. The reported optimizer is AdamW with initial learning rate $\phi_{t2v} = R_{t2v}(T,V)$9, and training runs for 120 epochs. The curvature parameter is selected through ablation, with $T_v = T \circ \phi_{t2v}.$0 chosen in the final setting (Li et al., 31 Jul 2025).

The experimental protocol also includes synthetic misalignment generation for originally aligned datasets. LLVIP is made misaligned by random nonlinear transformations applied to infrared images, MFNet by random nonlinear transformations applied to visible images, and RoadScene by random horizontal shifts of infrared images by 0.5–1.5% of image width. DroneVehicle is used as a real misaligned infrared–visible benchmark. Sobel edges are used throughout the adversarial and edge-level hyperbolic objectives (Li et al., 31 Jul 2025).

The method is therefore trained not merely as a fusion system, but as a coupled optimization in which alignment and fusion are co-determined. This design is explicit in the training algorithm, where deformation estimation, hyperbolic alignment, adversarial edge discrimination, and fusion are executed in one loop (Li et al., 31 Jul 2025).

6. Evaluation and empirical behavior

Hy‑CycleAlign is evaluated on DroneVehicle, LLVIP, MFNet, RoadScene, and TNO using registration metrics including HD, HD95, ASSD, DSC, and MEE, together with fusion metrics including SF and EN. The reported comparisons include alignment-free fusion methods such as DIDFuse, CDDFuse, and EMMA, and alignment-based methods such as SuperFusion, ReCoNet, MURF, UMF‑CMGR, and IMF (Li et al., 31 Jul 2025).

The ablation study on MFNet is especially diagnostic. A Euclidean baseline with cycle alignment but without H$T_v = T \circ \phi_{t2v}.$1CA reports HD $T_v = T \circ \phi_{t2v}.$2, HD95 $T_v = T \circ \phi_{t2v}.$3, ASSD $T_v = T \circ \phi_{t2v}.$4, DSC $T_v = T \circ \phi_{t2v}.$5, and EN $T_v = T \circ \phi_{t2v}.$6. The full Hy‑CycleAlign model reports HD $T_v = T \circ \phi_{t2v}.$7, HD95 $T_v = T \circ \phi_{t2v}.$8, ASSD $T_v = T \circ \phi_{t2v}.$9, DSC $\phi_{v2t} = R_{v2t}(T,V)$0, and EN $\phi_{v2t} = R_{v2t}(T,V)$1. Intermediate ablations show that H$\phi_{v2t} = R_{v2t}(T,V)$2CA‑p alone and H$\phi_{v2t} = R_{v2t}(T,V)$3CA‑e alone both improve performance, while their combination yields the strongest overall result (Li et al., 31 Jul 2025).

On DroneVehicle, the method reports HD $\phi_{v2t} = R_{v2t}(T,V)$4, HD95 $\phi_{v2t} = R_{v2t}(T,V)$5, ASSD $\phi_{v2t} = R_{v2t}(T,V)$6, DSC $\phi_{v2t} = R_{v2t}(T,V)$7, and MEE $\phi_{v2t} = R_{v2t}(T,V)$8, together with competitive SF and EN. On LLVIP it reports DSC $\phi_{v2t} = R_{v2t}(T,V)$9 and ASSD $V_t = V \circ \phi_{v2t}.$0. On MFNet it reports HD $V_t = V \circ \phi_{v2t}.$1, HD95 $V_t = V \circ \phi_{v2t}.$2, ASSD $V_t = V \circ \phi_{v2t}.$3, DSC $V_t = V \circ \phi_{v2t}.$4, and MEE $V_t = V \circ \phi_{v2t}.$5. These values are presented as either best or competitive relative to the listed baselines, depending on the dataset and metric (Li et al., 31 Jul 2025).

The curvature ablation compares $V_t = V \circ \phi_{v2t}.$6. The configuration $V_t = V \circ \phi_{v2t}.$7 yields HD95 $V_t = V \circ \phi_{v2t}.$8, ASSD $V_t = V \circ \phi_{v2t}.$9, DSC $$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$0, and EN $$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$1, and is adopted as the preferred trade-off. The complexity comparison reports 16.66 G FLOPs and 18.26 M parameters for Hy‑CycleAlign, versus 16.36 G FLOPs and 1.96 M parameters for SuperFusion, and much larger FLOP counts for MURF, UMF, and IMF (Li et al., 31 Jul 2025).

A downstream object-detection experiment on DroneVehicle uses YOLOv11‑m over fused images. Hy‑CycleAlign reports recall $$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$2, precision $$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$3, and mAP@0.5 $$T_{vt} = T_v \circ \phi_{v2t}, \qquad V_{tv} = V_t \circ \phi_{t2v}.$$4, with precision and [email protected] listed as best in that comparison. This result is consistent with the method’s central claim that registration quality directly affects fusion utility in downstream tasks (Li et al., 31 Jul 2025).

7. Relation to broader hyperbolic alignment research and open directions

Hy‑CycleAlign belongs to a wider class of methods that use hyperbolic geometry for cross-modal or structured alignment, but its task is notably different from most of that literature. Related work applies hyperbolic alignment to text–3D retrieval through hierarchical ordering and contribution-aware aggregation (Li et al., 14 Nov 2025), open-vocabulary semantic segmentation through decoupled radial and angular alignment (Truong et al., 9 May 2026), multi-modal entity alignment in knowledge graphs (Guo et al., 2021), vision–language dataset distillation through rank-aware hyperbolic alignment (Jeong et al., 28 Jun 2026), few-shot adaptation through path-decoupled hyperbolic flow matching (Li et al., 24 Feb 2026), brain–vision alignment through hyperbolic feature interpolation (Jo et al., 24 Mar 2026), manifold-preserving hyperbolic graph convolution (Dai et al., 2021), joint user–community network alignment (Sun et al., 2022), optimal-transport-based alignment of Poincaré embeddings (Hoyos-Idrobo, 2020), and modality alignment across heterogeneous hyperbolic manifolds with distinct curvatures (Wei et al., 31 Oct 2025). Within this landscape, Hy‑CycleAlign is distinguished by operating at the level of dense infrared–visible image registration and fusion rather than retrieval, segmentation, graph alignment, or representation adaptation (Li et al., 31 Jul 2025).

The paper also identifies several limitations and future directions. Hyperbolic operations increase parameter count and can be computationally heavier than purely Euclidean models. The current formulation uses a single Poincaré model, and more advanced or hybrid hyperbolic geometries are suggested for complex modality discrepancies. Additional directions named in the paper include broader modality pairs such as MRI–CT, PET–CT, LiDAR–camera, and SAR–optical, more robust handling of extreme misalignment through coarse-to-fine or hierarchical deformation models, and extensions toward real-time deployment and volumetric 3D registration (Li et al., 31 Jul 2025).